Slow-fast systems with stochastic resetting
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
In this paper we explore the effects of instantaneous stochastic resetting on a planar slow-fast dynamical system of the form ˙𝑥=𝑓(𝑥)−𝑦 and ˙𝑦=𝜖(𝑥−𝑦) with 0<𝜖≪1. We assume that only the fast variable 𝑥(𝑡) resets to its initial state 𝑥0 at a random sequence of times generated from a Poisson process of rate 𝑟. Fixing the slow variable, we determine the parameterized probability density 𝑝(𝑥,𝑡|𝑦), which is the solution to a modified Liouville equation. We then show how for 𝑟≫𝜖 the slow dynamics can be approximated by the averaged equation 𝑑𝑦/𝑑𝜏=𝔼[𝑥|𝑦]−𝑦 where 𝜏=𝜖𝑡, 𝔼[𝑥|𝑦]=∫𝑥𝑝∗(𝑥|𝑦)𝑑𝑥 and 𝑝∗(𝑥|𝑦)=lim𝑡→∞𝑝(𝑥,𝑡|𝑦). We illustrate the theory for 𝑓(𝑥) given by the cubic function of the FitzHugh–Nagumo equation. We find that the slow variable typically converges to an 𝑟-dependent fixed point 𝑦∗ that is a solution of the equation 𝑦∗=𝔼[𝑥|𝑦∗]. Finally, we numerically explore deviations from averaging theory when 𝑟=𝑂(𝜖).
Date Issued
2025-12-01
Date Acceptance
2025-07-21
Citation
SIAM Journal on Applied Dynamical Systems, 2025, 24 (4), pp.2549-2574
ISSN
1536-0040
Publisher
Society for Industrial and Applied Mathematics
Start Page
2549
End Page
2574
Journal / Book Title
SIAM Journal on Applied Dynamical Systems
Volume
24
Issue
4
Copyright Statement
Copyright © 2025 Society for Industrial and Applied Mathematics. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2025-10-06